polarization identity

Theorem [polarization identity] - Let $X$ be an inner product space over $ℝ$. The following identity holds for every $x,y\in X$:

$$⟨x,y⟩=\frac{1}{4}({\parallel x+y\parallel }^2-{\parallel x-y\parallel }^2)$$

If $X$ is an inner product space over $ℂ$ instead, the identity becomes

$$⟨x,y⟩=\frac{1}{4}({\parallel x+y\parallel }^2-{\parallel x-y\parallel }^2)+\frac{1}{4}i({\parallel x+iy\parallel }^2-{\parallel x-iy\parallel }^2)$$

Remark - This result shows that the inner product of $X$ is determined by the norm. Moreover, it can be shown that if a normed space $V$ the parallelogram law, the above formulas define an inner product compatible with the norm of $V$.